A mixture of two ideal gases with adiabatic exponents ${\gamma }_1$ and ${\gamma }_2$ and mole fractions $x_1$ and $x_2$ is kept in a container. What is the effective adiabatic exponent of the mixture?

Solution

1. The molar heat capacity at constant volume for a mixture is given by the weighted sum of the components: $C_{v,\text{mix}}=x_1{C}_{v1}+x_2{C}_{v2}$.
2. Using the relation $C_v=\frac{R}{\gamma -1}$, we substitute this into the mixture equation: $\frac{R}{{\gamma }_{\text{eff}}-1}=x_1\left(\frac{R}{{\gamma }_1-1}\right)+x_2\left(\frac{R}{{\gamma }_2-1}\right)$.
3. Dividing both sides by $R$ yields the final expression: $\frac{1}{{\gamma }_{\text{eff}}-1}=\frac{x_1}{{\gamma }_1-1}+\frac{x_2}{{\gamma }_2-1}$.
4. Hence the answer is (B).